How to make armature edit mode match object mode?
The change in location during render or closing/opening the file is due to the animation, which is taken in account during render and opening the file.

The procedure for having your shoot as rest position for both the mesh and the armature is this:

Select the body mesh and apply the armature modifier.
Do the same for the hands mesh.
Select ALL bones in pose mode and apply pose as rest pose.
Turn off the animation in the action editor.
Add an armature modifier to the body mesh, choosing your skeleton as "object".
Do the same for the hands mesh.